## The 61st Carnival of Mathematics

January 1st, 2010 | Categories: Carnival of Math, general math | Tags:

Today is a big day!  Not only is it the  first day of the new year but it’s also the first day of a new decade!  In addition to all of that it is also time for the 61st Carnival of Mathematics and this one has shaped up to be a great one thanks to the growing army of carnival contributors out there.  So, put off joining the gym for one more day; Sit back, relax and enjoy this feast of pulchritudinous mathematics.

First off, as per long standing carnival tradition, let’s look at some interesting properties of the number 61. Well, it’s prime for a start but so are a lot of numbers so maybe that isn’t so interesting. However, 61 is the smallest multidigit prime p such that the sum of digits of p^p is a square (pop-quiz – what is the next one?). While on the subject of primes, 61 is the smallest prime who’s digit reversal is square! It also turns out that the 61st Fibonacci number (2504730781961) is the smallest Fibonacci number which contains all the digits from 0 to 9. (Thanks to Number Gossip for these by the way).

So 61 is a lot more interesting than you thought huh?  If it had a name then it would be Keith and he would be Australian.

Puzzles, games and problems.

Let’s kick things off with a few puzzles.  Sam Shah has submitted a problem for you to try which was originally created by his sister (A physics teacher) in A stubborn equilateral triangle.

If you are in the market for some online math games and lessons then head over to TutorFi.com and see what Meaghan Montrose has found for you.

Jonathan has a very interesting puzzle over at his blog, jd2718,  called Who Am I (Teacher Edition) which should keep you thinking while you recover from the new year festivities.

Finally, Erich Friedman has prepared a set of holiday puzzles for 2009 for you all to try.

Explorations, discussions and messing about with maths

Pi is irrational right?  Have you ever seen the proof?  If not then you need to check out Brent Yorgey’s three part series on the irrationality of Pi over at The Math Less Travelled.  Part 3 of this series forms Brent’s submission to today’s carnival.

Pat Ballew has had his math class working on some maximization problems recently and his article Exploring an Isoperimetric Theme discusses a discovery about a “rule of thumb” for some maximization problems.  In a later post he wonders if there is a relation between the shape of a polygon and the maxium length of the diagonals (for a fixed perimeter).

Something that has kept potamologists awake over the years is the geometry of meandering rivers.  If you’ve ever wondered about the same thing then head over to Division by Zero to see what Dave Richeson has to say on the subject.

Terry Tao has been getting into the holiday season with a ‘more frivolous post than usual’ in A demonstration of the non-commutativity of the English language while Qiaochu Yuan of Annoying Precision gets more serious and considers the combinatorics of words in The cyclotomic identity and Lyndon words.

Matters of a statistical nature

John D. Cook has been contemplating questions involving rare diseases and counterfeit coins over at The Endeavour.

Over at An Ergodic Walk the author has been discussing a statistical problem of how to estimate a probability distribution from samples when you don’t know e.g. how many possible values there are.  An example application is estimating the number of different butterfly species from a sample containing many unique species.

I don’t know about you but I like to have a game of cards with my mates from time to time (Poker is our usual game of choice and I am fantastically bad at it).  Every now and then a small ‘discussion’ breaks out concerning how shuffled the deck of cards is which is usually solved by somone reshuffling them ‘properly’.  But how many suffles are necessary to randomize a deck of cards? Mathematically, card-shuffling can be viewed as a random walk on a finite group and, thus, it can be modeled by a Markov chain. Rod Carvalho has the details.

Techno Techno Techno….the technological side of mathematics

Sage is one of the best free mathematical software packages you can get at the moment and the project is led by William Stein, an associate professor at The University of Washington.  In his post Mathematical Software and Me he discusses his past experiences with mathematical software and recounts the series of events that led him to start the development of Sage.  If you are interested in Sage and can program in Python and Javascript then you may want to consider my Sage Bounty Hunt.

Wolfram Alpha has been a big hit among mathematics bloggers this year and, since it was launched back in May, Wolfram Research have added a lot of new features to it.  For a list of some of the more recent features check out the latest post from the Wolfram Alpha Blog – New Features in Wolfram|Alpha: Year-End Update.

Visualisation of volume data is getting easier every day thanks to products such as MATLAB and  Patrick Kalita recently gave an internal talk to Mathworks engineers explaining how to do it.    This talk was recorded and turned into a series of 9 blog posts by Doug Hull over at Doug’s MATLAB Video Tutorials and the final part was posted early in December.

Teaching, learning and testing

Explaining mathematics can be hard and there are many different ways of teaching it.  In How We Teach, Joel Feinstein shares some of his methodologies and includes a screencast of a talk of his entitled “Using a tablet PC and screencasts when teaching mathematics.”

Every year, many hundreds of mathematics graduate students take language exams where they translate some technical writing in French,German or Russian into English.  These translations are then graded and thrown away which seems like a waste of effort when you think about it.  David Speyer wonders if there is a more useful way to administer language exams in his post Let’s make language exams useful.

Finally, Eric Mazur has posted a ‘video confession’ on YouTube saying “I thought I was a good teacher until I discovered my students were just memorizing information rather than learning to understand the material. Who was to blame?”

Happy new Year – Math Carnival Style!

Now that it is officially 2010 you will be in need of a new calendar which is where Ron Doerfler of Dead Reckonings comes in.  He has created a great looking calendar called The Age of Graphical Computing and has made it all available for free. Just download, print and away you go.  I have to confess that I am too lazy to build them myself and so only wish that he could make them available for sale somehow.  What about it Ron?

So, that’s it for this edition of the carnival – I hope you enjoyed it.  The next one will be published on February 4th and I am still looking for someone to host it. So, if you have blog about mathematics and would like a traffic boost then drop me a line and we’ll discuss it.

1. Pub-quiz next primes are:
313, 463, 739, 37957, 54667, 66571, 80809, 98041…
I checked until 125003 (which has 637131 digits).

Mathematica code:
CheckNum[n_Integer]:=IntegerQ[Sqrt[Total[IntegerDigits[n^n]]]]
i=PrimePi[62];
Dynamic[i]
While[True,If[CheckNum[Prime[i]],Print[Prime[i]]];i++]

2. Great Site keep up the good work.

3. I really enjoyed the stubborn triangle question! Although it was solved in a couple of lines ;-)

{x,y}=.;
p=Table[{Cos[\[Theta]],Sin[\[Theta]]},{\[Theta],{0,2Pi/3,4Pi/3}}];
m={1,4,9};
forces=Total[MapThread[#2/Norm[{x,y}-#1]^3 ({x,y}-#1)&,{p,m}]];
forcemagnitude=forces.forces
sol1 = {x, y} /. NMinimize[{forcemagnitude, x^2 + y^2 < 1 \[And] x < 0 \[And] y > 0}, {x, y}, MaxIterations -> 10^6][[2]]
sol2 = {x, y} /. NMinimize[{forcemagnitude, x^2 + y^2 < 1 \[And] x > 0 \[And] y < 0}, {x, y},  MaxIterations -> 10^6][[2]]

After which you can plot:
ContourPlot[Log[forcemagnitude], {x, -1.1, 1.1}, {y, -1.1, 1.1}, PlotPoints -> 10, PlotRangePadding -> 0,
Contours -> 20, Epilog -> {Red, PointSize[Large], Point[{sol1, sol2}], Orange,Point[p]},
RegionFunction -> (#1^2 + #2^2 < 1 &), FrameLabel -> {"x", "y"}]


Solutions are (we can have two solutions because the fields decay with distance squared):
{-0.477646475033636079575008490756, 0.174709836993090262676835042780}
{0.601438719335870730289962396423, -0.116809119162900477536841860600}

4. Some more pub-quiz:

61
313
463
739
37957
54667
66571
80809
98041
152029
202087
362749
440269

the last one having 2484755 digits!